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In this work we propose and analyse a numerical method for computing a familyof highly oscillatory integrals with logarithmic singularities. For thesequadrature rules we derive error estimates in terms of $N$, the number ofnodes, $k$ the rate of oscillations and a Sobolev-like regularity of thefunction. We prove that that the method is not only robust but the error evendecreases, for fixed $N$, as $k$ increases. Practical issues about theimplementation of the rule are also covered in this paper by: (a) writing downready-to-implement algorithms; (b) analysing the numerical stability of thecomputations and (c) estimating the overall computational cost. We finish byshowing some numerical experiments which illustrate the theoretical resultspresented in this paper.

Filon-Clenshaw-Curtis rules for a class of highly-oscillatory integrals with logarithmic singularities